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Tuesday, January 14, 2014

So, I'm going to take a page out of my friend and colleague +Artem Kaznatcheev 's playbook and write a blog post about a project that I'm nearing the start of. My DPhil thesis is centered around the study of the 'cancer stem cell' hypothesis, and how it affects tumour progression. You might remember a post earlier about an agent based model we've built to study this, and they'll be more in the future, covering other aspects such as radiobiologic response and niche evolution. The first paper should be out soon in PLoS Computational Biology, and in the mean time there is a #preprint on the #bioRxiv here.

I've been slamming my head against the wall for the past several weeks working to write up the model in the form of a thesis chapter, which I'm finding is VERY different than writing a paper (at least for Oxford's Centre for Mathematical Biology). So, as I can't stand it any more, I'm going to write this post about what I'm planning to be the final research chapter in my thesis - an exploration of plasticity in the cancer stem-cell phenotype using Evolutionary Game Theory (EGT). EGT is a technique that has been used for half a century or so to study the evolutionary dynamics of populations containing species (or players) with different life-strategies (called payoffs). It differs from standard Game Theory in that players can't change strategies, but instead, their frequency in the population will change based on the relative fitness as governed by the replicator equation.

Phew - that was a long introduction. Anyways, I'm eager to do some EGT in my thesis, and no one has tried to make sense of cancer stem-cell plasticity with this technique, so I figure I'll give it a go. We're interested in what sort of conditions would result in promotion of the stem phenotype, why the stem fraction would be heterogeneous and how different sorts of stem-cell niches would affect this fraction. These are good kinds of questions to ask using EGT as the end result is (typically) ranges in parameter space that map to certain population proportions in the long run (called the Evolutionary Stable State which you can read about here).

So - what's the game then? Well, we've thought long and hard about how to structure this sort of game. We've gone back and forth thinking about pitting one stem cell against another, different types of tumours (with different stem parameters) against one another, but have recently settled on trying to pit the stem cells vs. plastic daughters vs. non-plastic daughters. We're going to consider some intriguing data from our collaborator +Anita Hjelmeland about the role of IL-6 in promoting the stem phenotype and try to make some sense of all of this! We begin by thinking about the allowable phenotypic transitions and population changes as stem cells either self-renew (probability s) or divide asymmetrically to form a non-stem daughter and maintain their population number. The plastic progenitors can also self-renew (probability a) or differentiate (d) or dedifferentiate back into stem cells (1-a-d). You can see a schematic of this in the figure below:

We decided to move away from the standard formulations of EGT in this respect, and we consider these sorts of divisions (ones that increase or decrease a population) as being fitness payoffs. And, as I said we'd try to consider the effects of IL-6 which +Anita Hjelmeland and crew wrote about, we've added in an asymmetric cost (c) and benefit (b). In their paper, they found that both stem and non-stem cells produced IL-6, but that only stem cells benefitted from its presence. So, our final payoff table looks something like this:

hmm... you can't really read that - but I can't NOT include a picture of the chalk board, so there it is. Here's the payoff table we think we're going to go with. Now listen, if you are an EGT nerd (I'm talking AT LEAST to you +Artem Kaznatcheev - PLEASE DO NOT ANALYZE THIS GAME, or if you do, keep it to yourself, I need a DPhil!).

Yes, I know there's a -c in every block and that I can simplify the game a bunch more. No one is sure if the cost of producing IL-6 (c) is the same across cell types, so we're still thinking on it.

So - that's where I'm going to start. With any luck we can learn something. Worst case, I'll be able to make some pretty pictures and do some proper analysis. Next post will be an analysis of the three 2x2 subgames, a la Artem's method (analyze, blog, analyze, blog, PAPER!).

More soon. If you have feedback on the payoff table, I'd LOVE to hear it (BEFORE I start analyzing it).

I approach the understanding of cancer like my original training in physics taught me - from the ground up, using the descriptive language of mathematics. Using established mathematics in new ways, guided by the principles of evolution, I hope to better understand (and maybe treat!) cancer. I am a proud member of the Integrated Mathematical Oncology group at the Moffitt Cancer Center and the Centre for Mathematical Biology at Oxford University.